Definition of Function

Q.

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Q. If $a+\alpha =1,b+\beta =2$ and $af\left(x\right)+\alpha f\left(\frac{1}{x}\right)=bx+\frac{\beta }{x},x\ne 0,$ then the value of the expression $\frac{f\left(x\right)+f\left(\frac{1}{x}\right)}{x+\frac{1}{x}}$ is

Q. Which among the following relations defined on $R$ is/are functions

1. $\left(i\right)$
2. $\left(ii\right)$
3. $\left(iv\right)$
4. $\left(v\right)$

Q. Which among the following is/are odd functions?

1. $g\left(x\right)=\frac{a^x-1}{a^x+1}$
2. $f\left(x\right)=x\left(\frac{e^x-1}{e^x+1}\right)$
3. $h\left(x\right)=\log (x^6+\sqrt{x^2+1})$
4. $k\left(x\right)=x(x^2-|x|)$

Q. If $f:N\to N$, where $N$ is set of natural numbers and $f\left(xy\right)=f\left(x\right)\cdot f\left(y\right)-f(x+y)+1\forall x,y\in N$ and $f\left(1\right)=2$, then the value of $1+\sum _{k=1}^{\infty }\frac{f\left(k\right)}{2^k}$ is

Q. Let $A$ and $B$ be two sets such that $n\left(A\right)=6$ and $n\left(B\right)=3,$ then number of onto functions from $A$ to $B$ is

Q. If $f:[\frac{1}{2},\infty )\to [\frac{3}{4},\infty )$ given by $f\left(x\right)=x^2-x+1,$ then $f^{-1}\left(x\right)=$

1. $\frac{1}{2}+\sqrt{x-\frac{1}{4}}$
2. $\frac{1}{4}+\sqrt{x-\frac{1}{2}}$
3. $\frac{1}{2}+\sqrt{x-\frac{3}{4}}$
4. $\frac{1}{2}-\sqrt{x-\frac{3}{4}}$

Q. Let $R$ be a relation on a set $A$ such that $R=R^{-1}$, then $R$ is

1. a reflexive relation
2. a symmetric relation
3. a transitive relation
4. None of the above

Q. Which of the following is not a function?

1. 
2. 
3. 
4. 

Q. If the ${10}^{th}$ term of an $A.P.$ is $35$ and the $5^{th}$ term is $20$, then

1. $a=9$
2. $d=3$
3. $a=8$
4. $T_9=32$

Q. Which among the following is\are function(s)

1. 
2. 
3. 
4. 

Q. Which of the following is/are functions?

1. 
2. 
3. 
4. 

Q. Which of the following is/are a function?

1. 
2. 
3. 
4. 

Q. Let $A=\{x_1,x_2,x_3,x_4,x_5,x_6\}$ and $f:A\to A$. The number of bijective functions such that $f\left(x_i\right)=x_i$ for exactly four of the $x_i$ is

Q. Which of the following functions is inverse to itself?

1. $f\left(x\right)=\frac{1-x}{1+x}$
2. $f\left(x\right)=5^{\log x}$
3. $f\left(x\right)=\sin x$
4. $f\left(x\right)=2^{x(x-1)}$

Q. If $n\left(A\right)=n,n>0$ then which among the following statements are correct

1. Number of relations on $A$ that are not reflexive $=\left(2^{n^2-n}\right)(2^n-1)$
2. Number of relations on $A$ that are not reflexive $=\left(2^{n^2+n}\right)(2^n-1)$
3. Number of relations on $A$ that are not symmetric
   $=2^{n^2}-2^{\frac{n(n+1)}{2}}$
4. Number of relations on $A$ that are not symmetric
   $=2^{n^2}-2^{\frac{n(n-1)}{2}}$